Journal of Hydrology 504 (2013) 132–141

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Monotonicity of the cell-centred triangular MPFA method for saturatedand unsaturated flow in heterogeneous porous media

0022-1694/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.jhydrol.2013.09.041

⇑ Corresponding author. Tel.: +33 (0)3 68 85 04 48; fax: +33 (0)3 68 85 04 02.E-mail address: fahs@unistra.fr (M. Fahs).

Anis Younes, Marwan Fahs ⇑, Benjamin BelfortLaboratoire d’Hydrologie et de Géochimie de Strasbourg, Université de Strasbourg/EOST, CNRS, UMR 7517, 1 rue Blessig, 67084 Strasbourg, France

a r t i c l e i n f o

Article history:Received 29 April 2013Received in revised form 16 September 2013Accepted 23 September 2013Available online 30 September 2013This manuscript was handled by PeterK. Kitanidis, Editor-in-Chief, with theassistance of Renduo Zhang, Associate Editor

Keywords:Saturated/unsaturated flowMPFA methodMonotonicityUpwind schemeNon-physical oscillations

s u m m a r y

In this work, the monotonicity of the symmetric cell-centred Multi-Point Flux Approximation (MPFA)method on triangular grids is studied for saturated and unsaturated flow in heterogeneous porous media.For the linear saturated flow, we derive necessary conditions on the triangulation to satisfy the PositiveTransmissibility (PT) condition which ensures monotonicity of the discrete scheme. For the nonlinearunsaturated flow, an upwind MPFA scheme is developed using the upstream weighting of the hydraulicconductivity. We show analytically that this scheme cannot satisfy the PT condition whatever the kind ofthe triangulation. In spite of that, numerical experiments in 2D heterogeneous domains and unstructuredmeshes show that the upwind MPFA scheme avoids spurious oscillations in the numerical solution.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Multi-Point Flux Approximation (MPFA) methods use more thantwo elements to compute fluxes across edges. The MPFA methodsare well suited for flow in saturated and unsaturated zones (Bauseand Hoffmann, 2008; Klausen et al., 2008) since they are locallyconservative and can handle general irregular grids on anisotropicand heterogeneous domains (Aavatsmark, 2002; Edwards, 2002).Many types of MPFA methods exist. In this work, we are concernedwith the symmetric MPFA O-method on triangles with polygonaldual mesh. This formulation has been shown to be equivalent tothe lowest order Mixed Finite Element (MFE) of Brezzi–Douglas–Marini with a specific quadrature rule (Aavatsmark et al., 2005;Klausen and Winther, 2006a,b; Wheeler and Yotov, 2006a,b). Con-trarily to quadrilateral meshes where a loss of convergence hasbeen observed for rough grids (Klausen and Winther, 2006b;Wheeler and Yotov, 2006a,b; Younes and Fontaine, 2008), themethod was shown to be robust in the case of triangular meshes(Friis et al., 2008; Klausen et al., 2008; Lipnikov et al., 2009b) sinceit is not deteriorated with unstructured grids (Younes and Fontaine,2008). However, the symmetric triangular MPFA method can gener-ate solutions with spurious oscillations. Such numerical difficulties

are common for finite element methods (Celia et al., 1990; Forsythand Kropinski, 1997; Forsyth et al., 1995; Ju and Kung, 1997;Karthikeyan et al., 2001; Milly, 1985; Pan et al., 1996), MFE (Belfortet al., 2009; Farthing et al., 2003; Younes et al., 2006), and MPFA(Edwards and Zheng, 2008, 2010; Eigestad et al., 2002; Keilegavlenand Aavatsmark, 2011; Nordbotten and Aavatsmark, 2005;Nordbotten et al., 2007). The problem of non-physical oscillationsis often related to the monotonicity of the scheme. Monotonicityof MPFA methods has been studied for quadrilateral grids inEdwards and Zheng (2008), Nordbotten and Aavatsmark (2005),Nordbotten and Eigestad (2005) and Nordbotten et al. (2007). TheM-matrix condition of MPFA has been discussed in Edwards andRogers (1994, 1998) and Edwards and Zheng (2008, 2010). Newschemes with improved monotonicity have been developed to pro-duce nonoscillatory solutions especially for anisotropic domains(Edwards and Zheng, 2010; Friis and Edwards, 2011; Le Potier,2005; Lipnikov et al., 2009a; Pal and Edwards, 2010).

Monotonicity of the triangular cell-centred MPFA formulationhas been investigated for saturated flow on uniform triangles andhomogeneous porous media in Keilegavlen and Aavatsmark(2011). The aim of this work is to study monotonicity of this for-mulation for both saturated and unsaturated flow on general trian-gular grids and heterogeneous domains.

In the first part of this paper, we consider the linear saturatedflow problem in heterogeneous domains. We derive, in this case,the necessary conditions on the triangulation to satisfy the Positive

A. Younes et al. / Journal of Hydrology 504 (2013) 132–141 133

Transmissibility (PT) condition which guaranties the monotonicityof the discrete scheme.

For the nonlinear unsaturated flow, the PT condition is not suf-ficient and should be combined with an upstream weighting forthe interblock hydraulic conductivity estimation to ensure themonotonicity of the scheme (Forsyth and Kropinski, 1997). Hence,in the second part of this paper, we investigate an upwind triangu-lar MPFA scheme based on the upstream weighting of the hydrau-lic conductivity. We show analytically that this scheme cannotsatisfy the PT condition whatever the kind of the triangulation. Inspite of that, numerical experiments in 2D homogeneous and het-erogeneous domains show that the upwind scheme avoids spuri-ous oscillations in the numerical solution even for highlyunstructured meshes.

The paper is organised as follows: Section 2 is devoted to a suc-cinct description of the saturated-unsaturated flow model and itsdiscretization with the triangular cell-centred MPFA method. InSection 3, the monotonicity of the method is studied for saturatedflow in heterogeneous domains. In Section 4, an upwind schemefor unsaturated flow is developed using the upstream weightingof the hydraulic conductivity for each half-edge flux. In this sectionwe show that the PT condition cannot be satisfied with the upwindscheme. In Section 5, numerical experiments are performed usingstandard and upwind MPFA schemes to illustrate the efficiencyof upwinding to avoid local maxima and minima. Finally, conclud-ing remarks are summarised in Section 6.

2. Saturated-unsaturated flow model and numerical resolutionwith the cell-centred MPFA method

2.1. The mathematical model for variably saturated flow in porousmedia

Subsurface water flow in saturated-unsaturated porous mediacan be described by Richards’ equation, formed by the mass con-servation equation and Darcy-Buckingham’s law. Hence, the fol-lowing pressure head form can be considered:

ðcðhÞ þ SsSWðhÞÞ@h@tþr � ð�KðhÞrHÞ ¼ r ð1Þ

where c(h) [L�1] is the specific moisture capacity (c = dh(h)/dh), H [L]and h [L] are respectively the hydraulic piezometric and pressurehead such as H = h + z, z [L] is the depth taken positive upward, Ss

[�] is the specific storage, SW [�] is the relative saturation of theaqueous phase (SW(h) = h(h)/hs), h(h) [L3 L�3] and hs [L3 L�3] arethe actual and saturated water contents respectively and r [T�1] isthe source-sink term. The water velocity, q [L T�1], is givenfrom the Darcy-Buckingham law:

q ¼ �KðhÞrH ð2Þ

where K(h) [L T�1] is the hydraulic conductivity as a function ofpressure head.

In the case of saturated flow (c = 0, SW = 1, K(h) = Ks) (Ks [L T�1]refers to the saturated conductivity), Eqs. (1) and (2) are linear.

Table 1Unsaturated hydraulic models.

References Formulation and condition Parameters

Haverkamp et al. (1977) Se(h) = l/(l + |h|n) h < 0Kr(h) = b/(b + |h|m)

van Genuchten (1980) Se(h) = 1/[1 + (a|h|)n]m h < 0

KrðSeÞ ¼ SLe½1� ð1� S1=m

e Þm�2

When the pressure head condition is not satisfied, Se = 1 and Kr = 1.

However, for unsaturated flow, Eqs. (1) and (2) should be comple-mented by specifying the hydraulic relationships between K, h andh. In this study, these interdependencies are characterised by themodel of Haverkamp (HV) (Haverkamp et al., 1977) or that of Mua-lem–van Genuchten (MVG) (Mualem, 1976; van Genuchten, 1980).The effective saturation Se(h) [�] and the hydraulic conductivityK(h) defined by,

SeðhÞ ¼hðhÞ � hr

hs � hrð3Þ

KðhÞ ¼ Ks � KrðhÞ ð4Þ

are expressed as function of the pressure head h in Table 1 for bothmodels. In Eqs. (3) and (4), hr [L3 L�3] is the residual water contentand Kr(h) [�] the relative conductivity.

The system (1)-(4) is solved with the corresponding initial andboundary conditions of Dirichlet or Neumann type. The domain isdiscretized with unstructured triangular meshes. These meshes aresuitable for practical problems with complex geometry and localmesh refinement.

2.2. Numerical discretization with the cell-centred MPFA method

The MPFA method has been used for modelling water flowthrough saturated (Aavatsmark et al., 1994; Edwards and Rogers,1994, 1998; Fontaine and Younes, 2010; Younes and Fontaine,2008) and unsaturated (Bause and Hoffmann, 2008; Klausenet al., 2008) porous media. In this section, we recall the main stagesof the triangular cell-centred MPFA method for solving variablysaturated flow in porous media.

The finite volume formulation of Eq. (1) over an interior ele-ment E1 is written as function of the hydraulic head H = h + z, asfollowing:

jE1j cE1 ðhE1 Þ þ Ss;E1 SWðhE1 Þ� � dHE1

dtþX

j

Q E1j ¼ jE1jrE1 ð5Þ

where |E1| is the area of E1, hE1 and HE1 are the hydraulic pressureand piezometric heads at the centre of E1, rE1 is the sink/source termover E1 and QE1

j are the fluxes across the edges of E1.The basic idea of the cell-centred MPFA method is to divide each

triangle into 3 subcells as in Fig. 1. Inside the subcell (O, F1, G, F2)formed by the corner O, the centre G and the midpoint edges F1

and F2, we assume linear variation of the head between HE1 , k1

and k2, the head values at respectively G and the two continuitypoints f1 and f2. The symmetry of the MPFA method is achievedwhen the continuity points are localised at Of1

OF1¼ Of2

OF2¼ 2

3. In this case

(O, f1, G, f2) is a parallelogram. Therefore, half-edge fluxes

Q 1O ¼

R F1O �KðhÞrH and Q2

O ¼R F2

O �KðhÞrH� �

, taken positive for

outflow, write (see Younes and Fontaine, 2008).

Q1O

Q2O

!¼ �9

2jE1j�OF1��!� K � Of1

�!OF1��!� K � f2O

�!OF2��!� K � Of1

�!OF2��!� K � f2O

�! !

k1 � HE1

k2 � HE1

� �ð6Þ

In the case of isotropic media, half-edge fluxes simplify to:

b, l: displacement pressuresn, m: parameters related to the uniformity of the pore size distribution

m = 1 � 1/n (traditionally)n and a: empirical constantsL: pore connectivity parameter(L = 0.5 following Mualem, 1976)

Fig. 1. Triangle splitting into three subcells and linear pressure approximation onthe subcell.

134 A. Younes et al. / Journal of Hydrology 504 (2013) 132–141

Q 1O

Q 2O

!¼ bE1

�OF1��! � OF1

��!OF1��! � OF2

��!OF1��!� OF2��!

�OF2��!� OF2��!

!k1 � HE1

k2 � HE1

� �ð7Þ

with bE1¼ 3KE1=jE1j .

The system (7) is written for all elements of the interaction re-gion formed by the subcells sharing the vertex O (see Fig. 2). Thenby writing continuity of fluxes across half-edges and continuity ofthe head at continuity points, we obtain a local system½A�ðkÞ ¼ ½B�ðHÞ which, in the case of the local numbering of Fig. 2,writes:

ðbE1þbE5

ÞOF1��!�OF1

��! �bE1OF1��!�OF2

��!0 0 �bE5

OF1��!�OF5

��!�bE1

OF2��!�OF1

��! ðbE2þbE1

ÞOF2��!�OF2

��! �bE2OF2��!�OF3

��!0 0

0 �bE2OF3��!�OF2

��! ðbE3þbE2

ÞOF3��!�OF3

��! �bE3OF3��!�OF4

��!0

0 0 �bE3OF4��!�OF3

��! ðbE4þbE3

ÞOF4��!�OF4

��! �bE4OF4��!�OF5

��!�bE5

OF5��!�OF1

��!0 0 �bE4

OF5��!�OF4

��! ðbE3þbE4

ÞOF5��!�OF5

��!

0BBBBBBBBB@

1CCCCCCCCCA

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}A

k1

k2

k3

k4

k5

0BBBBBB@

1CCCCCCA

¼

bE1F1O��!�F1F2

��!0 0 0 bE5

F1O��!� F1F5

��!bE1

F2O��!� F2F1��!

bE2F2O��!�F2F3��!

0 0 0

0 bE2F3O��!�F3F2

��!bE3

F3O��!�F3F4

��!0 0

0 0 bE3F4O��!�F4F3

��!bE4

F4O��!�F4F5

��!0

0 0 0 bE4F5O��!�F5F4

��!bE5

F5O��!�F5F1

��!

0BBBBBBBBB@

1CCCCCCCCCA

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

HE1

HE2

HE3

HE4

HE5

0BBBBBB@

1CCCCCCA

ð8Þ

Fig. 2. The interaction region sharing the vertex O.

This local system is inverted to obtain the head at the continuitypoint (ki) as function of the head at all elements sharing the vertexO:

ki ¼X

j

NijHEjð9Þ

where the matrix N is given by [N] = [A�1][B].Then, substituting Eq. (9) into the system (7), gives half-edge

fluxes explicitly as a weighted sum of the cell heads of the interac-tion volume. For example, the first flux Q1

O depicted in Fig. 2 writes:

Q1O¼bE1

�OF1��!�OF1��! X

j

N1jHEj�HE1

!þOF1��!�OF2��! X

j

N2jHEj�HE1

!" #

ð10Þ

The six half-edge fluxes (Q1O, Q2

O, Q1P , Q2

P , Q1R and Q2

R of Fig. 1) for theelement E1 are written in the same manner. The summation of thesefluxes is then substituted into Eq. (5) and leads to:

jE1j cE1 hE1

� �þ Ss;E1 SWðhE1 Þ

� �dHE1

dtþXnv

k¼1

dEkHEk¼ jE1jrE1 ð11Þ

where Ek represents all mesh elements sharing at least one node ofthe element E1, dEk

is the contribution of Ek and nv is the number ofEk.

The time integration of Eq. (11) is performed using higher ordertime integration methods with automatic time stepping schemesvia the DLSODIS (Double precision Livermore Solver for Computa-tional issues of hybrid or Ordinary Differential Equations – Implicitform and Sparse matrix) solver (Fahs et al., 2009; Hindmarsh,1980; Seager and Balsdon, 1982; Younes et al., 2009, 2011). Thesolver attempts to adapt simultaneously the time step size andthe order of integration to reach a prescribed accuracy.

3. Monotonicity of the cell-centred triangular MPFA method forsaturated flow

Monotonicity of numerical schemes is an important issue to en-sure that the computed pressure heads and water fluxes are inagreement with the physics. For linear problems, the PT conditionis sufficient to guaranty the monotonicity of the discrete scheme(Forsyth and Kropinski, 1997). Hence, in this section, we derivethe necessary conditions on the triangulation to satisfy the PT con-dition in the case of saturated flow in heterogeneous domains.

The PT condition can be easily obtained if the discrete equationis written in the following form:

jE1jSs;E1

dHE1

dt¼Xj2nv

c1;jðHEj� HE1 Þ þ jE1jrE1 ð12Þ

In that case, the PT condition writes:

c1;j P 0 ð13Þ

In the following, we provide the main stages to obtain the system(12) from the triangular symmetric MPFA method and then we dis-cuss requirements to satisfy the PT condition (Eq. (13)).

Without loss of generality, we use the local numbering given inFig. 2. In the case of weakly acute triangulation (all angles are lessthan p/2), all scalar products in the system (8) are positive. In thiscase, the local system [A](k) = [B](H) has the following properties:

(a) All diagonal terms of the matrix A are positive ðAii P 0Þ;(b) All off-diagonal terms are negative ðAij;j–i 6 0Þ;(c) the matrix A is strictly diagonally dominant (|Aii| >

Pj,j–i|Aij|).

Note that item (c) can be easily demonstrated, for instance, forthe first line of the matrix A in the system (8), as following:

A. Younes et al. / Journal of Hydrology 504 (2013) 132–141 135

jA11j �Xj;j–1

jA1jj ¼ ðbE1þ bE5

ÞOF1��!� OF1��!� bE1

OF1��!� OF2��!� bE5

OF1��!� OF5��!

¼ bE1F1O��! � F1F2

��!þ bE5F1O��! � F1F5

��!> 0

The previous properties imply that all terms of the inverse matrixare positive ðA�1

ij P 0Þ. Similarly, in the case of acute triangulation,all scalar products in the matrix B are positive and therefore allterms in B are positive (Bij P 0). Consequently, the terms of the ma-trix [N] = [A�1][B] satisfy

Nij P 0 ð14Þ

On the other hand, we can observe that for each row, the sum ofterms in the matrix A is equal to the sum of terms in the matrixB. Using the unit column vector notation ðuÞ ¼ ð1 1 1 1 1 ÞT ,we have:

½A�ðuÞ ¼ ½B�ðuÞ ð15Þ

Hence,

ðuÞ ¼ ½N�ðuÞ ð16Þ

This shows that the sum of terms of each row of the matrix [N]verifiesX

j

Nij ¼ 1 ð17Þ

The final discrete system can obtained by substituting the sixhalf-edge fluxes (Q1

O, Q2O, Q1

P , Q2P , Q1

R and Q2R of Fig. 1) into the finite

volume formulation, which leads to:

Fig. 3. Results for test case 1: comparison of water table position obtained

Table 2Material properties for the two test problems.

Media/Zone Model Ks (cm/s) hs (cm3 cm�3) hr (cm3

A (test 1) MVG 9.722 � 10�3 0.3 0.01B (test 2) HV 9.44 � 10�3 0.287 0.0751 (test 3) MVG 9.153 � 10�3 0.368 0.101972 (test 3) MVG 5.445 � 10�3 0.351 0.098493 (test 3) MVG 4.805 � 10�3 0.325 0.085894 (test 3) MVG 4.805 � 10�2 0.325 0.08589

jE1jSs;E1

dHE1

dt¼ �ðQ1

O þ Q2OÞ � ðQ

1P þ Q 2

PÞ � ðQ1R þ Q 2

RÞ

þ jE1jrE1 ð18Þ

Using the system (7) we have,

ðQ1OþQ2

OÞ¼ bE1ðOF1��!�OF2

��!�OF1��!�OF1

��!Þðk1�HE1 ÞþðOF1��!�OF2

��!�OF2��!�OF2

��!Þðk2�HE1 Þh i

ð19Þ

which writes using Eq. (17),

ðQ 1O þ Q 2

OÞ ¼bE1ðOF1��! � F1F2

��!Þ k1 �X

j

N1jHE1

!"

þOF2��!� F2F1��!

k2 �X

j

N2jHE1

!#ð20Þ

Then using Eq. (9), we obtain,

�ðQ1O þ Q2

OÞ ¼X

j

cO1;jðHEj

� HE1 Þ ð21Þ

with

cO1;j ¼ bE1

N1jðF1O��!� F1F2��!Þ þ N2jF2O

��!� F2F1��!h i

ð22Þ

Therefore, Eq. (18) for the element E1 writes

jE1jSs;E1

dHE1

dt¼X

j

cO1;jðHEj

� HE1 Þ þX

j

cP1;jðHEj

� HE1 Þ

þX

j

cR1;jðHEj

� HE1 Þ þ jE1jrE1 ð23Þ

with MPFA_up versus experimental measurements at different times.

cm�3) l (�) or a (cm�1) n (�) m (�) b (�)

0.033 4.1 – –1.611 � 106 3.96 4.74 1.175 � 106

28 0.0334 1.982 – –06 0.0363 1.632 – –75 0.0345 5 – –75 0.0345 5 – –

Fig. 4. The uniform acute triangular mesh obtained by subdivision of a slightlyinclined uniform rectangular grid.

Fig. 5. Water content distribution h (cm3 cm�3) for coarse and fine grids after11.5 days of infiltration (lengths in centimetres).

136 A. Younes et al. / Journal of Hydrology 504 (2013) 132–141

This equation corresponds to the desired form (Eq. (12)). Moreover,with an acute triangulation, the scalar products in Eq. (22) are po-sitive. Consequently, Eq. (14) leads to

cO;P;R1;j P 0 ð24Þ

which corresponds to the PT condition (Eq. (13)).In conclusion, the results of this section show that, in the case of

saturated flow, the symmetric MPFA method satisfies the PT condi-tion and is therefore monotone in the case of acute triangulationeven for heterogeneous domains.

Fig. 6. Water content profiles obtained with MPFA_std and MPFA_up using coarseand fine meshes after 11.5 days of infiltration.

4. Monotonicity of the cell-centred triangular MPFA method forunsaturated flow

In the case of nonlinear unsaturated flow, the PT condition is nomore sufficient and should be combined with the upstreamweighting of the interblock conductivity to produce monotonicphysical results (Diersch and Perrochet, 1999; Forsyth and Kropin-ski, 1997; Krabbenhøft, 2007).

A. Younes et al. / Journal of Hydrology 504 (2013) 132–141 137

With standard finite volume approaches, the flux across an edgeseparating two cells is only function of the pressure head in thesecells. Therefore, upwind formulation can be easily implemented toevaluate the interblock conductivity. With the MPFA method, morethan two elements are used for the approximation of the fluxes andconsequently upstream pressure cannot be obtained directly.

To obtain an upwind scheme for the triangular MPFA method,we separate the two half-edge fluxes Q 1

O and Q2O (see Fig. 1) of

the system (7) as following:

Q 1O ¼ b1

O �OF1��! � OF1

��!ðk1 � HE1 Þ þ OF1��! � OF2

��!ðk2 � HE1 Þh i

Q 2O ¼ b2

O OF1��! � OF2

��!ðk1 � HE1 Þ � OF2��! � OF2

��!ðk2 � HE1 Þh i ð25Þ

where b1O ¼ 3K1

O=jE1j and b2O ¼ 3K2

O=jE1j with K1O and K2

O the conduc-tivities associated to the half-edges (OF1) and (OF2). Then, the con-ductivity at each half-edge is calculated using the upstreampressure head value of the element sharing the common half-edge.Using the notations of Fig. 2, K1

O is for instance defined as follow:

K1O ¼

KðhE1 Þ if Q 1O > 0

KðhE5 Þ if Q 1O < 0

(ð26Þ

Fig. 7. Regions where the PT condition is violated with MPFA_up for coarse mesh(lengths in centimetres, t = 11.5 days).

Note that in this case, Eq. (19) becomes:

ðQ 1O þ Q 2

OÞ ¼ b2OOF1��!� OF2��!� b1

OOF1��!� OF1��!h i

ðk1 � HE1 Þ

þ b1OOF1��!� OF2��!� b2

OOF2��!� OF2��!h i

ðk2 � HE1 Þ ð27Þ

With b1O ¼ 3K1

O=jE1j and b2O ¼ 3K2

O=jE1j .Using Eq. (9), we obtain

�ðQ1O þ Q2

OÞ ¼X

j

cO;up1;j ðHEj

� HE1 Þ ð28Þ

with

cO;up1;j ¼ OF1

��!� ðb1

OOF1��!� b2

OOF2��!ÞN1j þ OF2

��!� ðb2

OOF2��!� b1

OOF1��!ÞN2j ð29Þ

As in the previous section, the final system is obtained by substitut-ing the six half-edge fluxes (Q1

O, Q2O, Q1

P , Q2P , Q1

R and Q2R of Fig. 1) into

the finite volume formulation (Eq. (18)).Note that cO;up

1;j in Eq. (29) can take negative values even for anacute triangulation since b1

O and b2O can have different values. This

shows that the PT condition cannot be satisfied with the upwindMPFA scheme contrarily to the standard scheme.

Fig. 8. Highly perturbed mesh with ei,1 and ei,2 in the interval [�0.3, 0.3] (lengths incentimetres).

Fig. 9. Regions where the PT condition is violated with perturbed mesh (lengths incentimetres, t = 11.5 days).

Fig. 10. Spatial domain and boundary conditions (all boundaries are imp

138 A. Younes et al. / Journal of Hydrology 504 (2013) 132–141

5. Validation of the upwind MPFA scheme: comparison withexperimental results

In this part, the developed upwind MPFA scheme is used for thesimulation of the variably saturated flow laboratory experiment ofVauclin et al. (1979). This experiment describes the transientwater-table position for a laboratory scale soil box under surfacerecharge conditions. The spatial domain is a rectangular soil slabof 600 cm � 200 cm dimensions with an initial horizontal table lo-cated at a height of 65 cm from the base. At the surface of the do-main, a constant flux (of 355 cm/day) is applied over a width of100 cm in the centre. Due to the symmetry, only the right handside of the domain is considered. Impervious boundary conditionsare imposed at all the boundaries except the vertical right handside of the initial water table where we impose a constant hydrau-lic head of 65 cm. Initially, the soil system is assumed to be athydrostatic equilibrium with respect to the water table. The MVGmodel is used to describe the constitutive relations between pres-sure head, water content and conductivity. The material properties(media A) are obtained from Clement et al. (1994) and given inTable 1. The storage coefficient Ss and the source/sink term are con-sidered to be zero. The problem is simulated using a triangularmesh of 2000 elements. The results are presented in Fig. 3 whichdepicts also the experimental results of Vauclin et al. (1979). Thisfigure shows an excellent agreement between MPFA and experi-mental results. This suggests the validity of the upstream MPFAscheme for modelling 2D variably saturated flow problems.

6. Monotonicity of MPFA for unsaturated flow

The lack of monotonicity for unsaturated flow, generally yieldsunphysical oscillations near sharp wetting fronts. These oscilla-tions can have a dramatic effect on the convergence of the numer-ical model especially at coarse grids. To study this effect, wesimulate in this section water infiltration into dry soils for bothhomogeneous and heterogeneous media. Two numerical models

ermeable except the infiltration zone at the top) for the test case 3.

A. Younes et al. / Journal of Hydrology 504 (2013) 132–141 139

are used for that purpose: (i) MPFA_std: based on the standardMPFA method (developed in Section 2) and (ii) MPFA_up: basedon the upwind MPFA scheme (developed in Section 4). We com-pare the results in terms of water content profiles and we checknumerically the PT condition.

6.1. Test case 2: infiltration into a homogeneous dry soil

This test case describes infiltration of water through a homoge-nous dry soil as in Baker et al. (1999). The spatial domain is a rect-angle of 4500 cm depth and 900 cm width. The initial pressurehead within the domain is h = �928.8 cm. A constant flux is appliedat the top of the domain (1.8 cm/day), while a constant pressurehead (�928.8 cm) is maintained at the bottom. The lateral sidesare assumed to be impervious. The constitutive relationships be-tween pressure head, hydraulic conductivity and water contentare described by the HV model (Table 1). The porous medium isa moderately to coarsely textured soil. The hydraulic parametersare listed in Table 2 (material B). The storage coefficient (Ss) andthe source/sink term are zero. The simulation is performed untila final time of 11.5 days.

6.1.1. Uniform triangular meshThe spatial discretization is first performed with uniform trian-

gular meshes, obtained from rectangular grids by subdividing eachrectangular cell into two triangles (Fig. 4). The initial rectangulargrids are slightly inclined by modifying node coordinates usingxi,new = xi,old + 10�4yi,old and yi,new = yi,old for all nodes to ensure acutetriangles. Coarse and fine meshes of respectively 600 and 1800 ele-ments are used for the simulation.

The spatial distributions of water content with MPFA_std andMPFA_up are shown in Fig. 5 for coarse and fine meshes. The resultsof MPFA_std on the coarse mesh show strong unphysical oscilla-tions near the wetting front. This phenomenon is clearly shownin Fig. 6 which depicts the vertical water content profile alongx = 450 cm. The unphysical oscillations disappear from the finemesh solution. On the other hand, MPFA_up yields non-oscillatorysolution with both coarse and fine meshes (Fig. 6). Due to upwind-

Fig. 11. Regions where the PT condition is violated for t

ing, the wetting front with MPFA_up is slightly more diffused thanwith MPFA_std. Close water content distributions are obtained withboth methods in the case of oscillation free solution (fine mesh re-sults). The obtained solution is validated against results of a veryfine mesh formed by 36,000 elements.

Fig. 7 depicts the regions where the PT condition is violatedwith MPFA_up on the coarse mesh at the end of the simulation. Thisfigure shows that the PT condition is essentially violated near thewetting front. Note that the PT condition is fully satisfied withMPFA_std since the triangulation is acute (as demonstrated inSection 3).

In conclusion, the results of the infiltration problem on uniformacute triangulations show that:

– The PT condition is not sufficient to guarantee monotonicity ofthe scheme in the case of unsaturated flow.

– The PT condition cannot be satisfied with MPFA_up even foracute triangulation.

– The upstream weighting of the hydraulic conductivity providesnonoscillatory solutions despite the violation of the PTcondition.

6.1.2. Perturbed meshUnstructured meshes are obtained by perturbing the nodes

(except boundary nodes) of the previous coarse mesh according to:

xi;new ¼ xi;old þ ei;1Dx and yi;new ¼ yi;old þ ei;2Dy

where Dx ¼ 150 cm and Dy = 90 cm are the dimensions of the initialrectangular cell (before subdivision into triangles), ei,1 and ei,2 arerandom numbers.

Many perturbed meshes are generated using different degreesof perturbation by changing the interval of variation of the randomnumbers ei,1 and ei,2. Fig. 8 shows the perturbed mesh for ei,1 and ei,2

in the interval [�0.3, 0.3]. With this mesh, the PT condition is vio-lated with both MPFA_up and MPFA-std (Fig. 9). With, MPFA_std, thePT condition is violated at obtuse triangles. With MPFA_up, it isviolated not only at obtuse triangles but also at acute trianglesespecially near the wetting front.

he test case 3 (lengths in centimetres, t = 30 days).

140 A. Younes et al. / Journal of Hydrology 504 (2013) 132–141

The results using perturbed meshes with different degrees ofperturbation show that:

– MPFA_std yields solutions with strong oscillations for allmeshes.

– MPFA_up provides oscillations-free solutions whatever thedegree of perturbation of the mesh.

Fig. 14. Saturation contours (0.4) for the test case 3: results of MPFA_up versusliterature results.

6.2. Test case 3: infiltration into a heterogeneous dry soil

This test case deals with infiltration into a heterogeneous drysoil (Diersch and Perrochet, 1999; Forsyth and Kropinski, 1997;Arico et al., 2012). The spatial domain is shown in Fig. 10. No flowboundary conditions are prescribed except the infiltration bound-ary section on the top. The initial pressure head is h = �734 cm.The MVG model is used for the hydraulic constitutive relationships(Table 1). Table 2 shows the material properties for each zone. Thestorage coefficient is neglected and there are no sink/source termsinside the domain. Water infiltration occurs during 30 days. The

Fig. 12. Saturation contours (0.3 and 0.4) for the test case 3: solution of MPFA_upwith 4 levels of mesh refinement.

Fig. 13. Saturation contours (0.3) for the test case 3: results of MPFA_up versusliterature results.

problem is highly nonlinear due to the very dry initial conditionsand the high value of the pore size index (n) for the zones 3 and 4.

Several simulations are performed using different heterogeneitycontrasts by decreasing the permeability of zones 3 and 4 by sev-eral orders of magnitudes. The results of simulations show that:

– Due to the strong non-physical oscillations, the MPFA_std failedto converge for almost all heterogeneity contrasts and for bothmeshes.

– With MPFA_up, the PT condition is violated especially near thewetting front and on the elements adjacent to the discontinu-ities of the permeability (Fig. 11).

– MPFA_up does not encounter any convergence difficulty andprovides oscillation-free solutions whatever the degree of het-erogeneity of the problem.

The MPFA_up results are then compared to those of Diersch andPerrochet (1999) and Arico et al. (2012). The problem is simulatedusing parameters of Table 2 on a coarse mesh (denoted mesh0)formed by 1250 elements and 670 nodes. As in Arico et al.(2012), this initial mesh is then progressively refined by dividingeach element into four triangles by connecting the midpoints ofthe triangle sides. Three levels of mesh refinement are consideredcorresponding to 5000 (mesh1), 20,000 (mesh2) and 80,000(mesh3) elements respectively. The saturation contours (0.3 and0.4) obtained on these meshes are depicted in Fig. 12. The resultsin this figure point out the robustness of the MPFA_up scheme sincethe contours obtained with the different mesh levels are very close.The MPFA_up solution (mesh3) is then compared to the results ofDiersch and Perrochet (1999) and that of Arico et al. (2012) inFigs. 13 and 14. A very good agreement can be observed betweenthe MPFA_up results and that of Diersch and Perrochet (1999).

7. Conclusions

Monotonicity of a numerical scheme is an important propertyto prevent internal oscillations without any physical meaning. Thispaper focused on the monotonicity of the cell-centred triangularMPFA O-method for saturated and unsaturated flow. In the firstpart, we studied the monotonicity of the method for saturated flowin heterogeneous domains. We demonstrated analytically thatweakly acute triangulation is a sufficient condition to ensure the

A. Younes et al. / Journal of Hydrology 504 (2013) 132–141 141

monotonicity of the symmetric MPFA method even for heteroge-neous domain.

For unsaturated flow, we implemented an upwind MPFAscheme by using upstream weighting of the hydraulic conductivityfor each half-edge. We showed analytically that, contrarily to thestandard MPFA scheme, the PT condition cannot be satisfied withthe upwind scheme even with acute triangulation.

Besides, for completing the analytical study, numerical investi-gations have been carried out in unsaturated conditions in order toobserve the monotonicity of the different MPFA schemes. Numer-ical experiments were performed to simulate 2D water infiltrationinto dry soils. The results of the simulations showed that: (i) thestandard MPFA scheme can generate strong unphysical oscillationsnear sharp wetting fronts even with acute triangulation and theseoscillations can have a dramatic effect on the convergence of themodel and (ii) although the PT condition cannot be satisfied withthe upwind scheme, the upstream weighting of the hydraulic con-ductivity was shown to be sufficient to produce physical mono-tonic solution for all performed experiments even with highlyunstructured meshes and/or highly heterogeneous domains.

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